Picard–Fuchs Equation Applied to Quadratic Isochronous Systems with Two Switching Lines

The present paper is devoted to study the problem of limit cycle bifurcations for nonsmooth integrable differential systems with two perpendicular switching lines. By using the Picard–Fuchs equation, we obtain the upper bounds of the number of limit cycles bifurcating from the period annuli of the quadratic isochronous systems [Formula: see text], [Formula: see text] and [Formula: see text], [Formula: see text], when they are perturbed inside a class of all discontinuous polynomial differential systems of degree [Formula: see text]. This method can be applied to study the limit cycle bifurcations of other integrable differential systems.

Isochronous dynamical systems

This is a terse review of recent results on isochronous dynamical systems, namely systems of (first-order, generally nonlinear) ordinary differential equations (ODEs) featuring an open set of initial data (which might coincide with the entire set of all initial data), from which emerge solutions all of which are completely periodic (i.e. periodic in all their components) with a fixed period (independent of the initial data, provided they are within the isochrony region). A leitmotif of this presentation is that ‘isochronous systems are not rare’. Indeed, it is shown how any (autonomous) dynamical system can be modified or extended so that the new (also autonomous) system thereby obtained is isochronous with an arbitrarily assigned period T , while its dynamics, over time intervals much shorter than the period T , mimics closely that of the original system, or even, over an arbitrarily large fraction of its period T , coincides exactly with that of the original system. It is pointed out that this fact raises the issue of developing criteria providing, for a dynamical system, some kind of measure associated with a finite time scale of the complexity of its behaviour (while the current, standard definitions of integrable versus chaotic dynamical systems are related to the behaviour of a system over infinite time).

ON PLANAR AND NON-PLANAR ISOCHRONOUS SYSTEMS AND POISSON STRUCTURES

We construct certain new classes of isochronous dynamical systems based on the recent constructions of Calogero and Leyvraz. We show how a Poisson structure can be ascribed to such equations in ℝ3 and indicate their connection with the Nambu structures.